College Algebra Exam Review 317 - n log 2 . dim K .L// C 1...

7.3. ADJOINING ALGEBRAIC ELEMENTS TO A FIELD 327 We have already observed that the set of algebraic numbers is count-able, so this set is a countable ﬁeld. n Exercises 7.3 7.3.1. Show that if K ± L are ﬁelds, then the identity of K is also the identity of L . Conclude that L is a vector space over K . 7.3.2. Fill in the details of the proof of 7.3.1 to show that f ± i ² j g spans M over K . 7.3.3. If K ± L is a ﬁnite ﬁeld extension, then there exist ﬁnitely many elements a 1 ;:::;a n 2 L such that L D K.a 1 ;:::;a n /: Give a different proof of this assertion as follows: If the assertion is false, show that there exists an inﬁnite sequence a 1 ;a 2 ;::: of elements of L such that K ² ¤ K.a 1 / ² ¤ K.a 1 ;a 2 / ² ¤ K.a 1 ;a 2 ;a 3 / ² ¤ :::: Show that this contradicts the ﬁniteness of dim K .L/ . 7.3.4. Suppose dim K .L/ < 1 . Show that there exists a natural number n such that

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